approximate conserved quantities of conservative

Quaestiones Mathematicae 28(2005), 305–315. c 2005 NISC Pty Ltd, www.nisc.co.za °

APPROXIMATE CONSERVED QUANTITIES OF CONSERVATIVE DYNAMICAL SYSTEMS IN R3 ¨ G. Unal Faculty of Sciences, Istanbul Technical University, Maslak, 80626, Istanbul, Turkey. E-Mail [email protected]

C.M. Khalique International Institute for Symmetry Analysis and Mathematical Modelling, Department of Mathematical Sciences, North West University, Mafikeng Campus, P. Bag X2046, Mmabatho 2735, South Africa. E-Mail [email protected]

Abstract. Approximate generalized symmetries associated with the resonances of conservative dynamical systems in R3 have been discussed. It has been shown that an approximate first integral (conserved quantities) can be found from the knowledge of two conservative approximate symmetries. Furthermore, the result has been applied to a physical system. Invariant curves have been obtained analytically and they have been compared with the numerical ones on the Poincar´e surface of section. Approximate conserved quantity pinpoints the type of stability.

Mathematics Subject Classification (2000): 22E70. Key words: Conservative dynamical systems, approximate symmetries and conserved quantities, resonances.

1. Introduction. The regular behaviour (order) of the trajectories observed in the numerical studies of the nearly integrable Hamiltonian systems [6], [7] pinpointed the existence of approximate conserved quantities (or first integrals). Numerous perturbation methods have been developed to construct approximate conserved quantities, e.g., direct method of Contopoulos [6] and Birkhoff-Gustavson normal form method [10]. A comprehensive study of these methods and others can be found in [10]. Yet, none of these methods resort to the celebrated Noether’s theorem (see for example [16]) which provides a link between the exact Noether symmetries of the dynamical systems and the exact conserved quantities. Recently, ¨ Unal [13] has shown that the extension of Noether’s theorem to approximate symmetries leads to approximate conserved quantities which are in good agreement with numerical results. Approximate symmetry groups of differential equations ¨ has been developed by Baikov et.al. [1]. Based on their definition, Unal has given a method in [13] which incorporates the resonance relations between the eigenvalues of the linear part. Therefore, our approach remains intact with normal form theory (see for instance [12]). Applications of approximate symmetries to galactic dynamics can be found in [14] and [15]. 305

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Here we consider conservative dynamical systems of the form x˙ = f0 (x) + εf1 (x),

x ∈ R3

(1)

where ε is a small positive perturbation parameter. We assume that the unperturbed part is linear, i.e., f0 (x) = Lx (2) where the matrix L has a semi-simple eigenvalue spectrum. The vector field associated with the conservative dynamical system (1) is F = F0 + εF1 ,

Fb =

3 X

fbl (x)

l=1

∂ , ∂xl

(b = 0, 1)

(3)

with the property ∇ · F = 0.

(4)

Following [13], we derive the determining system for approximate generalized symmetries. We prove that an approximate conserved quantity can be obtained from the knowledge of two approximate generalized symmetries. We also give an algorithm to construct approximate conserved quantities. As an application, we consider a conservative dynamical system        0 1 0 0 x˙ 1 x1  x˙ 2  =  0 0 1   x2  + ε  0  (5) x˙ 3 x3 0 −1 0 gx31 which models the motion of electrical motor [12]. This dynamical system has an equilibrium point located at the origin of the phase space. The linear part has eigenvalues of the form λ1 = 0, λ2 = i and λ3 = −i. This shows that the dynamical system (5) is already on the center manifold. Therefore, no further reduction of order is possible via the center manifold theorem [17]. Furthermore, linear stability analysis is insufficient to determine the stability of this system. We apply our method to determine the stability of this system. We determine two conservative approximate symmetries. Using the proposition given here we obtain an approximate conserved quantity. The level curves of the latter are not closed. Therefore, we conclude that it is unstable. Finally, we compare our result with the numerically obtained Poincar´e surface of section. 2. Approximate generalized symmetries and conserved quantities. Nonlinear differential equations usually do not admit nontrivial exact Lie symmetries which leave differential equations form invariant. Therefore, we will consider approximate symmetries in this work. Our approach will remain intact with the normal form theory [17]. Normal forms inherit the structural stability properties (bifurcations, etc.,) of the dynamical systems. The normal form equations of the autonomous systems are also autonomous [17]. When a dynamical system admits a nontrivial symmetry the normal form converges [5]. In [13], it has been shown

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that normal forms of autonomous systems are, indeed, the exact generalized symmetries of the linear (unperturbed) part of the dynamical system. Therefore, we confine ourselves to time-independent approximate symmetries of the autonomous dynamical systems. To achieve that we now relax exact invariance condition and look for transformations which leave the system invariant approximately. Let us consider x ¯k = xk + a(η0k (x) + εη1k (x) + O(ε2 )) + O(a2 ) (6) where a is a group parameter and ε is a small positive parameter. We now substitute (6) into (1) to obtain 3 µ X j=1

∂f k η0j 0 ∂xj

− f0j

∂η0k ∂xj

¶

+ε

µX 3 µ j=1

∂f k η1j 0 ∂xj

− f0j

∂η1k ∂η k ∂f k − f1j 0 + η0j 1 ∂xj ∂xj ∂xj

¶¶

= 0. (7)

The determining system of equations for the symmetries can be obtained by evaluating (7) in ascending order of ε. These are given by ¶ 3 µ k k X j ∂f0 j ∂η0 η0 − f0 = 0, ∂xj ∂xj j=1 3 µ X j=1

η1j

∂η k ∂f0k − f0j 1 ∂xj ∂xj

¶

=

¶ 3 µ X ∂η k ∂f k f1j 0 − η0j 1 . ∂xj ∂xj j=1

We have just proven the following proposition. Proposition 1. For X to be an approximate symmetry vector field of (1), it must satisfy [X, F] = O(ε2 ), (8) where, [, ] is the Lie bracket and X = X0 + εX1 ,

Xb =

3 X

ηbl (x)

l=1

∂ , ∂xl

(b = 0, 1).

Here, X is the first-order approximate symmetry vector field. As it is seen from the first set of partial differential equations in (7), X0 is the exact symmetry vector field of the linear part of the system corresponding to F0 . This set of equations can be solved by the method of characteristics. It can be verified that it forms an infinite dimensional Lie algebra. However, this does not give us further information regarding stability issues. In order to remain intact with the normal form theory ([13], [12]), we now follow a different avenue. Introducing the transformation

into (1), we obtain

x = Sz

(9)

z˙i = λi zi + εf¯i1 (z).

(10)

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The determining system of equations for (10) becomes 3 X j=1

3 X

λ j zj

∂ η¯0k − λk η¯0k = 0, ∂zj 3

λ j zj

j=1

X ∂ η¯1k − λk η¯1k = ∂zj j=1

µ

− f¯1j

¶ ∂ f¯k ∂ η¯0k + η¯0j 1 ∂zj ∂zj

(11)

(12)

where λ1 , λ2 and λ3 are eigenvalues of L. Solutions of (11) and (12) can be transformed to solutions of (7) via η0k =

3 X

S kl η¯0l (S−1 x, t) and η1k =

l=1

3 X

S kl η¯1l (S−1 x, t).

(13)

l=1

The set of partial differential equations (11) is called the homological equation in the context of normal form theory [12]. Its solutions in the space of homogenous polynomials read as [13]: X s s s j η¯0j = z j1 z2j2 z3j3 , C0s (14) j1 sj2 sj3 1 sj1 λ1 +sj2 λ2 +sj3 λ3 −λj =0

j where C0s are the group parameters, and the sum in (14) will be taken over j1 sj2 sj3 all the resonant monomials which satisfy the resonance conditions

sj1 λ1 + · · · + sj3 λ3 − λj = 0.

(15)

Substitution of (14) into (11) leads to the infinitesimals of the first-order approximate symmetry η¯1k =

X

sk1 ,sk2 ,sk3

Ksk1 sk2 sk3 z sk1 z sk2 z sk3 , sk1 λ1 + sk2 λ2 + sk3 λ3 − λk 1 2 3

(16)

for non-resonant monomials. Here, Ksk1 sk2 sk3 are the coefficients of the corresponding non-resonant monomials. We now are prepared to search for the connection between conservative approximate symmetries and conserved quantities. This will be provided in the next proposition. Proposition 2. Suppose that the conservative dynamical system given in equation (1) admits two linearly independent approximate divergence free symmetries of the form X1 = X10 + εX11 and X2 = X20 + εX21 . Approximate conserved quantity I = I0 + εI1 can be obtained from X2 cX1 cΩ = dI + O(ε2 ),

(17)

where c is the interior product, Ω = dx1 ∧ dx2 ∧ dx3 (∧ is the wedge product) and d is the exterior derivative.

Approximate conserved quantities of dynamical systems in R 3

Proof. It follows from direct calculation and Poincar´e’s lemma [16].

309

2

One can devise a practical algorithm to find approximate conserved quantities from formula (17). The left hand side of (17) gives X2 cX1 cΩ = X20 cX10 cΩ + ε(X20 cX11 cΩ + X21 cX20 cΩ) = dI0 + εdI1 . This leads to ∂I0 ∂I0 ∂I0 = η023 η012 − η022 η013 , = η021 η013 − η023 η011 , = η022 η011 − η021 η012 . ∂x1 ∂x2 ∂x3 ∂I1 = η023 η112 − η022 η113 + η123 η012 − η122 η013 , ∂x1 ∂I1 = η021 η113 − η023 η111 + η121 η013 − η123 η011 , ∂x2 ∂I1 = η022 η111 − η021 η112 + η122 η011 − η121 η012 . ∂x3

(18)

(19)

Integration of (18) and (19) yields the approximate first integral I = I0 + εI1 . 3. An application. Before we apply our results to the dynamical system given in (5), we would like to discuss the exact symmetries of this system. The dynamical system in (5) can be rewritten as a third-order ODE (Ordinary Differential Equation) u000 + u0 − gu3 = 0. (20)

We first checked whether the ODE in (20) passes the exact linearizability tests or not. The ODE in (20) cannot be linearized by point transformations according to linearizability criteria given in ([9]). It cannot be linearized either by a nonlocal transformation according to Berkovich’s criteria given in [2] and [3]. Therefore, we conclude that ODE in (20) does not admit seven point symmetries [11] and ten contact symmetries [8]. However, the ODE given in (20) admits translational symmetry ∂ X= . ∂t This allows us to reduce the order of (20) by one to obtain u3 d2 2 (y ) − 2g +2=0 2 du y

(21)

where y = du dt and u is the independent variable now. Following [4] we now introduce the new coordinates y2 u Q = 3 , T = 11/10 2 2 in (21) to obtain T3 d2 Q √ + gδ +1=0 (22) dT 2 Q

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where δ = ±1. It can be easily shown that Equation (22) arises from a Lagrangian p 1 dQ 2 ) − Q − 2δgT 3 Q. L= ( 2 dT One can study the asymptotic solutions of the ODE in (22) as it has been done in [4]. But we are interested in the behaviour of family of solutions of the ODE (20). Therefore, we now appeal to the results obtained in the previous section. Since the eigenvalues of the linear part of (5) are semi-simple, we can apply the results given in the previous section to the system in (5). The transformation given in (9) with   1 1 1 −i  S= 0 i 0 −1 −1 casts (5) into    0 z˙1  z˙2  =  0 0 z˙3

0 i 0

    0 z1 2(z1 + z2 + z3 )3 g 0   z2  + ε  −(z1 + z2 + z3 )3  . 2 z3 −i −(z1 + z2 + z3 )3

(23)

We next determine the resonant monomials since they form the generalized symmetries of the unperturbed part of (23). Resonant monomials of degree-one can be obtained by solving the algebraic equations (15) together with sj1 + sj2 + sj3 = 1. Using (14), we obtain ¯ 1 = iz2 ∂ − iz3 ∂ . X 0 ∂z2 ∂z3

(24)

¯ 1 into the right hand side of (11), we notice that no resonant After substitution of X 0 monomials appear. Therefore, we now resort to (16), to obtain ¯ 1 = g((3z 2 z2 + 3z 2 z3 + 3z1 z 2 + 3z1 z 2 + z 3 + 3z 2 z3 + z2 z 2 + z 3 ) ∂ X 3 3 2 2 3 2 1 1 1 ∂z1 ∂ 1 3 − (z1 + 3z12 z3 + 3z1 z22 + 6z1 z2 z3 + 3z1 z32 + z23 + 3z2 z32 + z33 ) 2 ∂z2 1 3 3 ∂ 2 3 2 2 2 − (z1 + 3z1 z2 + 3z1 z2 + 6z1 z2 z3 + 3z1 z3 + z2 + 3z2 z3 + z3 ) ). 2 ∂z3

(25)

Similarly, resonant monomials of degree-three can be obtained from the solutions of the algebraic equations sk1 λ1 + sk2 λ2 + sk3 λ3 − λk = 0,

sk1 + sk2 + sk3 = 3

(26)

which lead to ¯ 22 = z1 z2 z3 ∂ , ¯ 21 = z 3 ∂ , X X 0 1 0 ∂z1 ∂z1 ∂ 24 25 ¯ = z2z3 ¯ = z 2 z3 ∂ , X , X 2 0 1 0 ∂z2 ∂z3

¯ 23 = z 2 z2 ∂ , X 1 0 ∂z2 26 ¯ = z 2 z2 ∂ . X 3 0 ∂z3

(27)

Approximate conserved quantities of dynamical systems in R 3

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We now consider the linear combination of the Noether symmetries given in (27), i.e., 6 X ¯2 = ¯ 2k . X qk X 0 0 k=1

¯ 1 into the right hand side of (11), we notice that the following After substituting X 0 resonant monomials appear. r12 = (−24q1 + 10q2 + 12q3 + 12q5 )z13 z2 z3 + (6q2 + 12q4 + 12q6 )z1 z22 z32 , r22 = (−6q1 − 4q3 )z14 z2 + (−6q2 − 27q3 + 6q4 − 3q5 )z12 z22 z3 + (3q4 − 3q6 )z23 z32 ,

r32 = (−6q1 − 4q5 )z14 z3 + (−6q2 − 3q3 − 27q5 + 6q6 )z12 z2 z32 + (−3q4 + 3q6 )z22 z33 .

In order to be able to find η¯1k in the space of homogeneous polynomials, we now have to kill these resonant monomials by setting their coefficients equal to zero. We obtain the following equations − 24q1 + 10q2 + 12q3 + 12q5 = 0, 6q2 + 12q4 + 12q6 = 0, 3q1 + 2q3 = 0, − 6q2 + 6q4 − 3q5 − 27q3 = 0, q4 − q6 = 0, 3q1 − 2q5 = 0, − 6q2 + 6q6 − 3q3 − 27q5 = 0. Solutions read as q2 = 6q1 ,

3 q 3 = − q1 , 2

3 q 4 = − q1 , 2

q5 = q 3 ,

q6 = q 4 .

This, indeed, indicates that the symmetry-breaking occurs [13]. Formula given in (16) leads to ¯ 2 = ig (4(18z 4 z2 − 18z 4 z3 + 12z 3 z 2 − 12z 3 z 2 − 2z 2 z 3 − 72z 2 z 2 z3 (28) X 1 2 1 2 1 3 1 1 2 1 1 16 + 72z12 z2 z32 + 2z12 z33 − 3z1 z24 − 6z1 z23 z3 + 6z1 z2 z33 + 3z1 z34 + 14z24 z3 ∂ + 126z23 z32 − 126z22 z33 − 14z2 z34 ) + (−36z15 − 24z14 z3 − 144z13 z22 ∂z1 + 4z13 z32 − 66z12 z23 − 54z12 z2 z32 + 6z12 z33 − 4z1 z24 + 72z1 z23 z3 − 180z1 z22 z32 ∂ − 32z1 z2 z33 + 3z25 + 12z24 z3 + 12z22 z33 + 3z2 z34 ) + (36z15 + 24z14 z2 ∂z2 − 4z13 z22 + 144z13 z32 − 6z12 z23 + 54z12 z22 z3 + 66z12 z33 + 32z1 z23 z3 ∂ ). + 180z1 z22 z32 − 72z1 z2 z33 + 4z1 z34 − 3z24 z3 − 12z23 z32 − 12z2 z34 − 3z35 ) ∂z3 Invoking transformation (13) for (24)–(28) we obtain the following conservative approximate symmetries in terms of original variables x1 , x2 and x3 . X10 = x2

∂ ∂ ∂ + x3 − x2 , ∂x1 ∂x2 ∂x3

(29)

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g ∂ X11 = − (8x31 + 36x21 x3 + 12x1 x22 + 60x1 x23 + 15x22 x3 + 35x33 ) 8 ∂x1 ∂ 3g 2 2 2 + x2 (4x1 + 8x1 x3 + x2 + 5x3 ) 8 ∂x2 g ∂ + (8x31 + 12x21 x3 + 24x1 x23 + 3x22 x3 + 15x33 ) . 8 ∂x3

(30)

Hence, the first conservative approximate symmetry vector field will read X 1 = X10 + εX11 . Similarly, 1 ∂ (8x31 + 36x21 x3 + 12x1 x22 + 60x1 x23 + 15x22 x3 + 35x33 ) 8 ∂x1 ∂ 3 − x2 (4x21 + 8x1 x3 + x22 + 5x23 ) 8 ∂x2 ∂ 3 − x3 (4x21 + 8x1 x3 + x22 + 5x23 ) , 8 ∂x3

X20 =

g x2 (768x41 + 3872x31 x3 − 308x21 x22 + 6060x21 x23 − 840x1 x22 x3 128 ∂ + 3480x1 x33 + 215x42 + 10x22 x23 + 875x43 ) ∂x1 g 3 2 4 5 + (576x1 + 2688x1 x3 − 560x1 x2 + 5552x31 x23 − 1428x21 x22 x3 + 6060x21 x33 128 ∂ + 224x1 x42 − 840x1 x22 x23 + 3480x1 x43 + 215x42 x3 + 10x22 x33 + 875x53 ) ∂x2 g 4 3 2 2 2 2 2 + x2 (−192x1 − 1952x1 x3 − 252x1 x2 − 4380x1 x3 − 280x1 x2 x3 128 ∂ − 3480x1 x33 + 9x42 − 10x22 x23 − 875x43 ) . ∂x3

X21 =

The second conservative approximate symmetries will read as X2 = X20 + εX21 . Since the conservative system given in (5) admits two conservative approximate symmetries we now appeal to Proposition 2 to construct an approximate conserved quantity. To achieve this goal we integrate (18) and (19). This leads to g 1 (x1 + x3 )(x22 + x23 )(4x21 + 3x22 + 8x1 x3 + 7x23 ) + ε (2x72 8 8 − x1 x2 (8x51 + 36x41 x3 + 14x42 x3 + 35x21 x3 (−x22 + x23 )

I=

(31)

+ x31 (−8x22 + 60x23 ) + 7x1 (x42 − 5x22 x23 ))).

Let us consider the level curves of the approximate conserved quantity (31) on the x1 x2 plane. The level curves, are indeed, invariant under the flow operator generated by (3). Therefore, they are called invariant curves. It is clear that open invariant curves are the manifestation of unboundedness of the solutions.

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Figure 1: Analytical versus numerical invariant curves for g = 1 and ² = 0.1. a) Contour plot obtained from Equation (31) by setting x3 = 0. b) Poincare surface of section obtained from numerical integration of Equation (5).

0

-0.2

-0.4

-0.6

-0.8

-1 -1

-0.5

0 aL

-0.75 -0.5 -0.25

0 bL

0.5

1

0

-0.2

-0.4

-0.6

-0.8

0.25

0.5

0.75

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¨ G. Unal and C.M. Khalique

Therefore, open invariant curves indicate that the system is unstable. In Figure 1a) invariant curves have been plotted as contour lines of (31) for x3 = 0, g = 1 and ² = 0.1 . Since all the invariant curves are open we conclude that the system (5) is unstable. To verify this statement we compare it with numerical experiments. This necessitates the construction of Poincar´e surface of section numerically. To achieve this goal, dynamical system in (5) has been integrated numerically by employing fourth-order Runge-Kutta for a set of initial conditions chosen from Figure 1a). Every time the numerically obtained trajectory hit the x1 x2 plane we record these points to display on Figure 1b). Brief comparison of Figure 1a) and Figure 1b) indicates that the system (5) is, indeed, unstable in the neighborhood of the origin where the equilibrium point is located. ¨ Acknowledgement. G. Unal would like to thank C.M. Khalique for his kind hospitality during his visit to the International Institute for Symmetry Analysis and Mathematical Modelling, North West University, Mafikeng Campus where most of this work was done. References 1. V.A. Baikov, R.K. Gazizov and N.H. Ibragimov, Approximate symmetries, Math. USSR Sbornik 64 (1989), 427–441. 2. L.M. Berkovich, The method of an exact linearization of an n-order ordinary differential equations, Nonlinear Mathematical Physics 3 (1996), 341–350. 3.

, Transformations of ordinary differential equations, Proceedings of Institute of Mathematics of NAS of Ukraine 30 (2000), 25–34.

4. S. Bouquet, Hamiltonian structure and integrability of the stationary KuramotoSivashinsky equation, J. Math. Phys. 36 (1995), 1242–1258. 5. A.D. Bruno and S. Walcher, Symmetries and convergence of normal forms, J. Math. Anal. and Appl. 183 (1994), 571–576. 6. G. Contopoulos, On the existence of a third integral of motion , Astron. J. 68 (1962), 1–14. 7. M. Henon, and C. Heiles, The applicability of the third integral of motion, some numerical experiments, Astron. J. 69 (1964), 73–79. 8. N.H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations, Volume 1, CRC Press, Baton Roca, p. 28, 1994. 9. N.H. Ibragimov and S.V. Meleshko, Linearizations of third-order equations, Archives of ALGA 1 (2004), 71–92. 10. A.J. Lichtenberg and M.A. Lieberman, Regular and Stochastic motion, SpringerVerlag, New York, 1992. 11. F. Mahomed and P.G.L. Leach, Symmetry Lie algebras of nth order ODE, J. Math. Anal. Appl. 151 (1990), 80–107. 12. V.M. Starzhinskii, Applied methods in the theory of nonlinear oscillations, Mir, Moscow, 1980. ¨ 13. G. Unal, Approximate generalized symmetries, normal forms and approximate first integrals, Phys. Lett. A 269 (2000), 13–30.

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¨ 14. G. Unal and G. Gorali, Approximate First Integrals of a Galaxy Model, Nonlinear Dynamics, 28 (2002), 195–211. ¨ 15. G. Unal and C.M. Khalique, Approximate First Integrals of the H´enon-Heiles system revisited, Communications in Nonlinear Science and Numerical Simulation 10 (2005), 73–83. 16. C. von Westenholz, Differential forms in mathematical physics, North-Holland, Amsterdam, 1981. 17. S. Wiggins, Introduction to Applied Dynamical Systems and Chaos, Springer-Verlag, Berlin, 1990. Received 2 February, 2004 and in revised form 16 February, 2005.